Consider the set of acts generated from all (Anscombe-Aumann) mixtures of two acts f and g, \left\{ \alpha f+(1-\alpha)g:\alpha\in[0,1]\right\}, and think of choosing from this set. We propose a preference condition, monotonicity in mixtures, which says that clearly improving the act f (in the sense of weak dominance) makes putting more weight on f more desirable. We show that this property has strong implications for preferences exhibiting behavior as in the classic Ellsberg (1961) paradoxes. For example, we show that maxmin expected utility (MEU) preferences (Gilboa and Schmeidler 1989) satisfy monotonicity in mixtures if and only if the set of probability measures appearing in the preference representation is a singleton set. In other words, for MEU preferences, monotonicity in mixtures and Ellsberg behavior are incompatible. MEU is not the only class of preferences for which this incompatibility holds, and we explore several directions in which this stark finding may be extended, including to any preferences that rely on kinks to generate Ellsberg behavior. Moreover, we demonstrate that the incompatibility is not between monotonicity in mixtures and Ellsberg behavior (or even ambiguity aversion more generally) per se by showing compatibility between these properties for other models of preferences. For example, we show that smooth ambiguity preferences (Klibanoff, Marinacci and Mukerji 2005) can satisfy both properties simultaneously as long as they exhibit a coefficient of relative ambiguity aversion everywhere less than one. This last result is closely related to findings from earlier literature on comparative statics of risky portfolios under expected utility.